Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with differential-topology
Search options answers only
not deleted
user 8103
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
2
votes
manifolds with unusual rational cohomology rings
I'm not sure if this is quite what you're looking for, because:
(a) it's a $3$-manifold with boundary, and
(b) it's not the ring structure which is interesting;
but the Borromean Rings link compl …
- 35.9k
2
votes
Topological relationships between family of transversal intersections of manifolds
I believe they are homeomorphic.
Let $j: M\hookrightarrow\mathbb{R}^n$ denote the embedding of $M$. Your path defines a smooth homotopy $h: M\times [0,1]\to \mathbb{R}^n$ given by $h(x,t) = j(x)+a(t) …
- 35.9k
3
votes
Accepted
Is L-S category estimating the number of components of the critical set?
Doesn't the height function on a torus laying down on its side give a counter-exsmple? There are two circles of critical points, but $\mathrm{cat}(T^2) =3$.
Added: A constant function on a connected …
- 35.9k
3
votes
Cohomology classes represented by submanifolds
This is an answer to your more general question about how to define $$f_\ast:H^\ast(X;\mathbb{Z})\to H^{\ast+\dim(Y)-\dim(X)}(Y;\mathbb{Z})$$
when $f: X\to Y$ is a proper oriented map.
For $\ell$ suf …
- 35.9k
3
votes
Accepted
Fréchet manifold structure of C(M, N)
See Theorem III.1.11 on page 76 of "Stable mappings and their singularities" by Golubitsky and Guillemin for a proof that $C^\infty(M,N)$ is a Fréchet manifold when $M$ is compact.
- 35.9k
19
votes
Accepted
homology classes as immersed submanifolds
It might be better to split the question into 2 cases and 2 steps.
Step 1: Which homology classes in $X$ can be represented by continuous maps of closed smooth manifolds (ie, which classes are $f_*[M …
- 35.9k
3
votes
How does all of the bundles over a certain manifold characterize the homotopy class of the b...
Let me attempt to summarise some basic facts.
Let $G$ be a Lie (say) group, and let's let $M$ and $N$ be smooth compact manifolds as in the question. A principal $G$-bundle on $N$ is a bundle with s …
- 35.9k
2
votes
When is the union of embedded smooth manifolds a smooth manifold?
For simplicity, I will assume that all manifolds are connected and closed.
An obvious sufficient condition is that the for each pair of embeddings, their images are either disjoint or equal.
To see …
- 35.9k
6
votes
Accepted
perturbing one map to be transverse to a second map
I'm not sure what exactly you mean by perturb, but you can always make $f$ transverse to $g$ by a homotopy and this is often enough. This is proved in Section IV.2 of Kosinski's "Differential Manifold …
- 35.9k
3
votes
Accepted
Equivariant Whitney approximation
This is Corollary 1.12 in
Wasserman, A. G., Equivariant differential topology, Topology 8, 127-150 (1969). ZBL0215.24702.
The proof is essentially the same as the one given by Peter Michor in his answ …
- 35.9k
34
votes
Accepted
Is differential topology a dying field?
Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.
- 35.9k
8
votes
Dimension in Whitney's theorem
The following paper of Elmer Rees addresses precisely your question for Lie groups:
Rees, Elmer
Some embeddings of Lie groups in Euclidean space.
Mathematika 18 (1971), 152–156.
The main result is …
- 35.9k
1
vote
Connected representant of a framed cobordism class (reference needed)
The claim doesn't seem to be true when $n=0$ and $M^m$ is orientable.
In that case framed cobordism classes correspond to homotopy classes of maps $f:M^m\to S^m$, which by the Hopf degree theorem are …
- 35.9k
3
votes
Wilking's connectivity theorem
This is a consequence of Poincaré duality applied to the open $n$-manifold $X-Y$.
In particular, we have $H^i_c(X-Y)\cong H_{n-i}(X-Y)$ for all $i$, where $H^\ast_c$ denotes cohomology with compact s …
5
votes
Reference request: an elementary proof of Brouwer fixed-point theorem.
In these notes by Tony Carbery, it is mentioned that a proof along these lines appears in the book Differential Forms and Applications by do Carmo, where it is attributed to E. Lima.
- 35.9k